If they had all demanded independence there probably would have been bloodshed. Also there are benefits to not going independent and it’s not certain how long the North being independent will work out or how it will go down. They’ll likely need to make political ties and trade and commerce with the 6 kingdoms anyway.
Other kingdoms deciding to go independent is definitely something for a sequel or for fans to think about happening or being discussed in the future though. Just because something didn’t happen in the last episode doesn’t mean it isn’t ever going to happen. That’s why stories leave some things open for the audience to speculate on. Personally I figure they will all end up gaining some independence and the North will eventually move to more of a semi-autonomous status instead after struggling on their own for some time.
Thinking about it some more. It would have been cool for Tyrion or someone in the council to mention that the Dornish, etc were complaining and that something would have to be done about it and maybe suggest adding Dornish dignitaries to the council or some some other possible solution. I don't think it was necessary, but it would have been an interesting touch.

They don’t care who rules, as long as there’s stability and they have some influence. Letting the common people decide was laughable to some of them and was not a real option at this point. Tyrion’s thing about who has the best story doesn’t really make sense (though saying the one who knows the most stories and understands history the best makes the best ruler is an interesting thought.) but I don’t think they cared.
I also wondered if it’s some sort of ‘writers are cool and should have more respect and power’ bit by GRRM or D&D, with Bran standing in for writers/storytellers.

Democracy was never a realistic option, at this point. Bran being king is setting it up so that the various powerful lords can use political influence to decide who the next king is, thus giving them a say in the process. At the moment Tyrion and the council are the true rulers though, which is pretty interesting and also is a way for the various lords to exert their influence into policy. Keep in mind that there are still spots open on the council.